Question: Let $\mathbf{a} = \begin{pmatrix} 1 \\ -2 \\ -5 \end{pmatrix},$ $\mathbf{b} = \begin{pmatrix} \sqrt{7} \\ 4 \\ -1 \end{pmatrix},$ and $\mathbf{c} = \begin{pmatrix} 13 \\ -4 \\ 17 \end{pmatrix}.$  Find the angle between the vectors $\mathbf{a}$ and $(\mathbf{a} \cdot \mathbf{c}) \mathbf{b} - (\mathbf{a} \cdot \mathbf{b}) \mathbf{c},$ in degrees.
Answer: Note that the dot product of $\mathbf{a}$ and $(\mathbf{a} \cdot \mathbf{c}) \mathbf{b} - (\mathbf{a} \cdot \mathbf{b}) \mathbf{c}$ is
\[\mathbf{a} \cdot [(\mathbf{a} \cdot \mathbf{c}) \mathbf{b} - (\mathbf{a} \cdot \mathbf{b}) \mathbf{c}] = (\mathbf{a} \cdot \mathbf{c}) (\mathbf{a} \cdot \mathbf{b}) - (\mathbf{a} \cdot \mathbf{b}) (\mathbf{a} \cdot \mathbf{c}) = 0.\]Therefore, the angle between the vectors is $\boxed{90^\circ}.$